OFFSET
1,1
COMMENTS
P(2) = 5 is the only prime pentagonal number, all other factor as P(k) = (k/2)*(3*k-1) or k*((3*k-1)/2) and thus have at least 2 prime factors. P(k) is semiprime iff [k prime and (3*k-1)/2 prime] or [k/2 prime and 3*k-1 prime].
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Pentagonal Number.
FORMULA
EXAMPLE
a(1) = 12 because P(12) = A000326(12) = 12*(3*12-1)/2 = 210 = 2 * 3 * 5 * 7 is a 4-almost prime (in fact the primorial prime(4)#).
a(3) = 16 because P(16) = 16*(3*16-1)/2 = 376 = 2^3 * 47 is a 4-almost prime (the prime factors need not be distinct).
MATHEMATICA
Select[Range[400], PrimeOmega[PolygonalNumber[5, #]] == 4 &] (* Amiram Eldar, Oct 06 2024 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 14 2006
EXTENSIONS
82 inserted by R. J. Mathar, Dec 22 2010
STATUS
approved